Note that the operator \(A\)  generates a \(C_0\) -semigroup \((\mathrm {e}^{t A})_{t\ge 0}\)  which is analytic of angle \(\frac{\pi }{2}\)  on \(H\) , see [1]}. Moreover, there exists an orthonormal basis \(\left\lbrace \varphi _{n}\right\rbrace _{n \in \mathbb {N}}\)  in \(H\)  of eigenvectors of \(-A\)  such that \(\varphi _{n} \in D(A),\left\Vert \varphi _{n}\right\Vert =1\)  and \(-A \varphi _{n}=\lambda _{n} \varphi _{n}, \forall n \in \mathbb {N}\) , where \(\left\lbrace \lambda _{n}\right\rbrace _{n \in \mathbb {N}} \subset \mathbb {R}\)  denote the corresponding eigenvalues which can be ordered so that
\(-\kappa \le \lambda _1 \le \cdots \le \lambda _m \le 0 < \lambda _{m+1} \le \cdots \, \rightarrow \infty .\) 
